Problem 69 - Elementary and Intermediate Algebra

Question

Use the distributive law to factor each of the following. Check by multiplying. $$ 7+7 y $$

Step-by-Step Solution

Answer

\[ 7(1 + y) \]

Step 1 - Identify the common factor

Examine the terms in the expression to identify the common factor. The terms are 7 and 7y. Notice both terms include the number 7.

Step 2 - Factor out the common factor

Since 7 is the common factor, factor 7 out of each term. This means rewriting the expression as: \[ 7(1) + 7(y) \].

Step 3 - Simplify the inside terms

Simplify the terms inside the parentheses to get: \[ 7(1 + y) \]. This is the factored form of the expression.

Step 4 - Check by multiplying

To verify the factorization, distribute 7 back into the terms inside the parentheses: \[ 7 \times 1 + 7 \times y = 7 + 7y \]. The original expression is obtained, confirming the factorization is correct.

Key Concepts

factoring expressions

Factoring expressions might sound complicated, but it’s simply about recognizing patterns and breaking down expressions into simpler parts. When you factor an expression, you are looking for a way to express it as a product of simpler expressions. Think about how you might break down a number into its prime factors; it's a similar process.

Let's consider the example from the exercise: 7+7y. We notice both terms share a common factor – the number 7. By factoring out this common factor, we can rewrite the expression more compactly.

This process makes it easier to work with algebraic expressions, especially when solving equations or simplifying expressions.

common factor

Identifying a common factor is a crucial step in factoring expressions. A common factor is a number or variable that divides each term in the expression without leaving a remainder.

In the expression 7 + 7y, both terms (7 and 7y) include the number 7. Therefore, 7 is the common factor.

To factor out the common factor, you'd rewrite each term as a product involving the common factor. For instance, 7 + 7y becomes 7(1 + y) after factoring out 7.

By doing this, you simplify the expression, which can make solving equations or performing further algebraic operations significantly easier.

simplifying algebraic expressions

Simplifying algebraic expressions involves reducing them to their simplest form. This can involve factoring, combining like terms, and performing arithmetic operations.

Consider our example, 7 + 7y. By factoring out the common factor, we first rewrite the expression as 7(1 + y).

Next, we simplify inside the parentheses, by ensuring there are no further operations or common factors to reduce. In this case, 1 + y is already as simple as it gets.

To verify our work, we distribute the factor back through the parentheses: 7 * 1 + 7 * y = 7 + 7y, proving our simplification is correct. This step shows that simplifying can often make expressions more manageable and easier to work with in further algebraic computations.